Constructing Quadrilaterals

Draw perpendicular and parallel lines with a set square and a ruler, construct parallelograms, rhombuses, rectangles, squares, trapeziums, kites and other quadrilaterals from given data, and name the shape you have constructed.

Before you start

To construct a figure is to draw it accurately with instruments: a ruler, compasses, a protractor and a set square. Always sketch it first, freehand, with the given lengths and angles marked, and leave your construction arcs showing (Constructing Triangles).
A triangle with three given sides is constructed with compasses: draw one side, then draw an arc from each end with the other two lengths as radii. The arcs cross at the third vertex (Constructing Triangles).
Measure lengths to the nearest 0.1 cm and angles to the nearest degree. Before you measure the length or angle you are asked for, check yourself: measure one of the given lengths or angles again. If it is wrong, the rest will be too.
An expected answer on this page is given with a range: your measurement is accurate if it is within 0.2 cm of a length, or within $2^{\circ}$ of an angle, given here.
The properties you will build with (Properties of Special Quadrilaterals): opposite sides of a parallelogram are parallel and equal; the diagonals of a rhombus bisect each other at right angles; the line of symmetry of a kite cuts the other diagonal in half at right angles.
The drawings on this page are not to the real size: do not measure the screen. Construct on paper and measure your own drawing.

Perpendicular and Parallel Lines with a Set Square

A set square has a right angle. Held against a ruler and slid along it, it moves without turning, so each of its edges keeps the same direction. That is all you need to draw a line perpendicular to a given line, or parallel to it, through any point.

Example 1: The perpendicular from a point to a line

Draw a line $AB$ 8 cm long. Mark a point $P$ so that $AP = 5$ cm and $\angle PAB = 50^{\circ}$ . Construct the perpendicular from $P$ to $AB$ , meeting $AB$ at $N$ . Measure $PN$ .

Solution

Step 1. Draw $AB = 8$ cm with a ruler.
Step 2. Put the centre of the protractor on $A$ , with its base line along $AB$ , and mark $50^{\circ}$ . Draw a line from $A$ through the mark, and mark $P$ on it, 5 cm from $A$ .
Step 3. Lay a ruler along $AB$ . Put the set square against the ruler with one edge of its right angle along the ruler, and slide it along the ruler until the other edge of the right angle passes through $P$ .
Step 4. Hold the set square still and draw along its edge from $P$ to $AB$ . Mark $N$ , where the line meets $AB$ , and the right angle. Check yourself: $AP$ should still measure 5 cm. Then measure $PN$ .

$PN = 3.8$ cm (accurate if from 3.6 cm to 4.0 cm).

The perpendicular from a point to a line

The perpendicular from $P$ to the line $AB$ is the line from $P$ that meets $AB$ at right angles, at a point $N$ called the foot of the perpendicular.

$PN$ is the shortest distance from $P$ to the line $AB$ . It is called the distance from $P$ to $AB$ .

With a set square, the edge along the ruler and the edge you draw along are the two edges of its right angle. The same steps draw a line perpendicular to $AB$ through a point on $AB$ : slide the set square until its edge passes through that point.

Drawing the line through $P$ parallel to $AB$

Step 1. Place the set square with one edge exactly along $AB$ .
Step 2. Place a ruler firmly against another edge of the set square.
Step 3. Hold the ruler still. Slide the set square along the ruler, without turning it, until the edge that was on $AB$ passes through $P$ .
Step 4. Draw along that edge through $P$ . This line is parallel to $AB$ . Mark the two lines with arrows.

Why the line is parallel

The set square slides along a straight ruler without turning, so the edge you draw along makes the same angle with the ruler in its new place as it did on $AB$ .

The ruler is a transversal of $AB$ and the new line, and these two equal angles are corresponding angles. When corresponding angles are equal, the lines are parallel (corr. ∠s equal), as in the lesson Parallel Lines and Transversals.

Parallelograms, Rhombuses, Rectangles and Squares

To construct a special quadrilateral, draw what is given, then use its properties to find the corners that are not given. Each step is something you already know how to do: a length with a ruler or compasses, an angle with a protractor, a parallel or a perpendicular line with a set square.

Example 2: A parallelogram from two sides and the angle between them

Construct the parallelogram $ABCD$ with $AB = 6$ cm, $AD = 4$ cm and $\angle DAB = 65^{\circ}$ . Measure the diagonal $BD$ .

Solution

Step 1. Sketch the parallelogram and mark the given lengths and angle. In a parallelogram, $DC$ is parallel to $AB$ and $BC$ is parallel to $AD$ .
Step 2. Draw $AB = 6$ cm.
Step 3. At $A$ , measure $\angle DAB = 65^{\circ}$ with the protractor and draw the line. Mark $D$ on it, 4 cm from $A$ .
Step 4. With the set square and ruler, draw the line through $D$ parallel to $AB$ .
Step 5. In the same way, draw the line through $B$ parallel to $AD$ . The two lines meet at $C$ .
Step 6. Check yourself: $DC$ should measure 6 cm and $BC$ 4 cm, as opposite sides of a parallelogram are equal. Then join $BD$ and measure it.

$BD = 5.6$ cm (accurate if from 5.4 cm to 5.8 cm).

Another way to find C: with compasses

Opposite sides of a parallelogram are equal, so $C$ is 6 cm from $D$ (as $DC = AB$ ) and 4 cm from $B$ (as $BC = AD$ ).

After Step 3, set the compasses to 6 cm and draw an arc with centre $D$ ; set them to 4 cm and draw an arc with centre $B$ . The arcs cross at $C$ .

The two arcs cross at only one point on this side of $BD$ , and the corner $C$ found with the set square lies on both arcs. So the compasses find the same point $C$ .

Example 3: A rhombus from its diagonals

Construct the rhombus $ABCD$ whose diagonals are $AC = 8$ cm and $BD = 6$ cm. Measure the side $AB$ and $\angle DAB$ .

Solution

The diagonals of a rhombus bisect each other (diags. of //gram bisect each other) at right angles (diags. of rhombus are ⊥). So they cross at $O$ , the midpoint of each, with $AO = OC = 4$ cm, $BO = OD = 3$ cm and a right angle at $O$ .

Step 1. Draw $AC = 8$ cm. Measure 4 cm from $A$ and mark the midpoint $O$ .
Step 2. With the set square and ruler, draw the line through $O$ perpendicular to $AC$ .
Step 3. Set the compasses to 3 cm. With centre $O$ , draw arcs cutting the perpendicular on both sides of $AC$ , at $B$ and at $D$ .
Step 4. Join $AB$ , $BC$ , $CD$ and $DA$ . Check yourself: $BD$ should measure 6 cm. Then measure $AB$ and $\angle DAB$ .

Each side is the hypotenuse of a right-angled triangle whose other two sides are 4 cm and 3 cm, so all four sides come out equal, as they must in a rhombus.

$AB = 5.0$ cm (accurate if from 4.8 cm to 5.2 cm) and $\angle DAB = 74^{\circ}$ (accurate if from $72^{\circ}$ to $76^{\circ}$ ).

A rhombus given its side and one angle is constructed like the parallelogram of Example 2, with all four sides the same length (Practice 3).

Example 4: A rectangle

Construct the rectangle $PQRS$ with $PQ = 7$ cm and $QR = 4$ cm. Measure the diagonal $PR$ .

Solution

Step 1. Draw $PQ = 7$ cm.
Step 2. With the set square and ruler, draw lines perpendicular to $PQ$ at $P$ and at $Q$ , on the same side of $PQ$ . (A protractor set at $90^{\circ}$ does the same.)
Step 3. Set the compasses to 4 cm. With centre $P$ , then centre $Q$ , draw arcs cutting these lines at $S$ and $R$ .
Step 4. Join $SR$ . Check yourself: $SR$ should measure 7 cm. Then join $PR$ and measure it.

$PR = 8.1$ cm (accurate if from 7.9 cm to 8.3 cm).

A square is constructed in the same way, with all four sides the given length. A square can also be constructed from its diagonal (Practice 4).

Trapeziums and Kites

A trapezium has exactly one pair of parallel sides: draw one of them, then the other through the end of a slanting side, with the set square. A kite is easiest from its diagonals, because its line of symmetry cuts the other diagonal in half at right angles.

Example 5: A trapezium

Construct the trapezium $ABCD$ with $AB \parallel DC$ , $AB = 8$ cm, $AD = 4$ cm, $\angle DAB = 60^{\circ}$ and $DC = 5$ cm. Measure $BC$ and $\angle ABC$ .

Solution

Step 1. Draw $AB = 8$ cm.
Step 2. At $A$ , measure $60^{\circ}$ and draw the line. Mark $D$ on it, 4 cm from $A$ .
Step 3. With the set square and ruler, draw the line through $D$ parallel to $AB$ .
Step 4. Set the compasses to 5 cm. With centre $D$ , draw an arc cutting the parallel line on the same side as $B$ . This is $C$ .
Step 5. Join $BC$ . Check yourself: $\angle DAB$ should measure $60^{\circ}$ and $AD$ 4 cm. Then measure $BC$ and $\angle ABC$ .

$BC = 3.6$ cm (accurate if from 3.4 cm to 3.8 cm) and $\angle ABC = 74^{\circ}$ (accurate if from $72^{\circ}$ to $76^{\circ}$ ).

Example 6: A kite from its diagonals

A kite $ABCD$ has $AB = AD$ and $CB = CD$ . Its diagonals are $AC = 9$ cm and $BD = 6$ cm, and they cross at $O$ with $AO = 2$ cm. Construct the kite and measure $CB$ and $\angle ABC$ .

Solution

The diagonal $AC$ joins the vertices where the equal sides meet, so it is the line of symmetry of the kite: it cuts $BD$ in half at right angles (prop. of kite). So $BO = OD = 3$ cm and $\angle AOD = 90^{\circ}$ .

Step 1. Draw $AC = 9$ cm. Mark $O$ on it, 2 cm from $A$ .
Step 2. With the set square and ruler, draw the line through $O$ perpendicular to $AC$ .
Step 3. Set the compasses to 3 cm. With centre $O$ , draw arcs cutting the perpendicular at $B$ and at $D$ .
Step 4. Join $AB$ , $BC$ , $CD$ and $DA$ . Check yourself: $BD$ should measure 6 cm. Then measure $CB$ and $\angle ABC$ .

$CB = 7.6$ cm (accurate if from 7.4 cm to 7.8 cm) and $\angle ABC = 100^{\circ}$ (accurate if from $98^{\circ}$ to $102^{\circ}$ ).

Any Quadrilateral: Two Triangles

A diagonal splits a quadrilateral into two triangles. If enough is given to construct each of them, construct one triangle (Constructing Triangles), then the other on the other side of the diagonal.

Example 7: Four sides and a diagonal

Construct the convex quadrilateral $ABCD$ with $AB = 6$ cm, $BC = 5$ cm, $CD = 4.5$ cm, $DA = 4$ cm and the diagonal $AC = 7$ cm. Measure the diagonal $BD$ .

Solution

Step 1. Sketch $ABCD$ with the diagonal $AC$ . It splits the quadrilateral into $\triangle ABC$ , with sides 6 cm, 5 cm and 7 cm, and $\triangle ACD$ , with sides 4 cm, 4.5 cm and 7 cm.
Step 2. Draw $AB = 6$ cm.
Step 3. Construct $\triangle ABC$ from its three sides: an arc of radius 7 cm with centre $A$ and an arc of radius 5 cm with centre $B$ cross at $C$ . Join $AC$ and $BC$ .
Step 4. Construct $\triangle ACD$ on the other side of $AC$ from $B$ , so that $ABCD$ is convex: an arc of radius 4 cm with centre $A$ and an arc of radius 4.5 cm with centre $C$ cross at $D$ .
Step 5. Join $AD$ and $CD$ . Check yourself: $CD$ should measure 4.5 cm. Then join $BD$ and measure it.

$BD = 6.7$ cm (accurate if from 6.5 cm to 6.9 cm).

Why four sides are not enough

Four sides alone do not fix a quadrilateral. The diamonds of a folding gate keep the same sides, yet they change shape as the gate opens and closes.

The two quadrilaterals below both have $AB = 6$ cm, $BC = 5$ cm, $CD = 4.5$ cm and $DA = 4$ cm, but their diagonals $AC$ are different. A fifth fact, such as a diagonal or an angle, fixes the shape of a convex quadrilateral.

AC = 7

AC = 8

A quadrilateral can also be fixed by two sides and two angles at the ends of a third side: draw that side, both angles, then the two sides, and join the ends (Practice 9). Once a quadrilateral is constructed, you can measure it and name it.

Example 8: Construct, then name

Construct a quadrilateral $PQRS$ whose diagonals $PR$ and $QS$ are both 7 cm long, bisect each other at $O$ , and make $\angle POQ = 50^{\circ}$ . Measure its sides and angles, and name the quadrilateral.

Solution

Step 1. Draw $PR = 7$ cm and mark its midpoint $O$ , 3.5 cm from $P$ .
Step 2. At $O$ , measure $\angle POQ = 50^{\circ}$ and draw a line through $O$ , on both sides of $PR$ .
Step 3. Set the compasses to 3.5 cm. With centre $O$ , draw arcs cutting this line at $Q$ , and at $S$ on the other side of $O$ .
Step 4. Join $PQ$ , $QR$ , $RS$ and $SP$ . Check yourself: $QS$ should measure 7 cm. Then measure the sides and the angles.

Measured: $PQ = RS = 3.0$ cm, $QR = SP = 6.3$ cm, and each of the four angles is $90^{\circ}$ .

$\angle QPS + \angle PSR = 90^{\circ} + 90^{\circ} = 180^{\circ}$ , so $PQ \parallel SR$ (int. ∠s supp.). $\angle SPQ + \angle PQR = 90^{\circ} + 90^{\circ} = 180^{\circ}$ , so $PS \parallel QR$ (int. ∠s supp.). So $PQRS$ is a parallelogram, and a parallelogram with four right angles is a rectangle.

A square has four equal sides, but these sides are 3.0 cm and 6.3 cm, so $PQRS$ is not a square. The most specific name is a rectangle.

$PQRS$ is a rectangle: $PQ = RS = 3.0$ cm, $QR = SP = 6.3$ cm (each accurate within 0.2 cm), and every angle is $90^{\circ}$ .

Common mistakes

Rubbing out the arcs. The arcs show how each point was found. Leave them showing.
Letting the set square turn or the ruler move while drawing a parallel line. Hold the ruler still and slide the set square along it; if either turns, the new line is not parallel.
Guessing a midpoint. In Examples 3 and 8 the diagonals cross at their midpoints: measure half the length from one end.
Putting the crossing point of a kite's diagonals in the middle of its line of symmetry. The line of symmetry cuts the other diagonal in half; where it is cut itself is given in the question (in Example 6, 2 cm from $A$ ).
Constructing the second triangle on the same side of the diagonal as the first. In Example 7, $D$ must be on the other side of $AC$ from $B$ . On the same side, the arcs cross inside $\triangle ABC$ , and you get a quadrilateral that is not convex (an arrowhead shape, with a corner pointing inwards), not the one asked for.
Reading the wrong scale of the protractor. For an angle of $65^{\circ}$ the other scale gives $115^{\circ}$ (as $180^{\circ} - 65^{\circ} = 115^{\circ}$ ). An acute angle must look acute, and an obtuse angle obtuse.

Practice

For each question: sketch the figure with the given data, construct it on paper with your instruments, leaving the arcs showing, check yourself as the question says, and then measure.

A measurement within 0.2 cm of the length, or within $2^{\circ}$ of the angle, in the answer is accurate.

Practice 1

Draw a line $QR$ 9 cm long and mark a point $P$ with $QP = 6$ cm and $\angle PQR = 35^{\circ}$ . Construct the perpendicular from $P$ to $QR$ , meeting $QR$ at $N$ . Check yourself: measure $QP$ again. Then measure $PN$ , the distance from $P$ to $QR$ .

ShowHide solution

Step 1. Draw $QR = 9$ cm. At $Q$ measure $35^{\circ}$ , draw the line, and mark $P$ on it 6 cm from $Q$ .
Step 2. Lay a ruler along $QR$ and put one edge of the set square's right angle along the ruler. Slide the set square until its other edge passes through $P$ .
Step 3. Draw from $P$ along the edge to $QR$ , and mark $N$ and the right angle. Check $QP = 6$ cm, then measure $PN$ .

$PN = 3.4$ cm (accurate if from 3.2 cm to 3.6 cm).

Practice 2

Construct the parallelogram $PQRS$ with $PQ = 7$ cm, $PS = 5$ cm and $\angle SPQ = 50^{\circ}$ . Check yourself: $SR$ should measure 7 cm. Then measure the diagonal $QS$ .

ShowHide solution

Step 1. Draw $PQ = 7$ cm. At $P$ measure $50^{\circ}$ , draw the line and mark $S$ on it, 5 cm from $P$ .
Step 2. Opposite sides of a parallelogram are equal: $SR = PQ = 7$ cm and $QR = PS = 5$ cm. Draw an arc of radius 7 cm with centre $S$ and an arc of radius 5 cm with centre $Q$ ; on the other side of $QS$ from $P$ , they cross at $R$ . (Or draw the lines through $S$ parallel to $PQ$ and through $Q$ parallel to $PS$ with the set square.)
Step 3. Join $SR$ and $QR$ . Check $SR = 7$ cm, then join $QS$ and measure it.

$QS = 5.4$ cm (accurate if from 5.2 cm to 5.6 cm).

Practice 3

Construct the rhombus $ABCD$ with sides of 5 cm and $\angle DAB = 70^{\circ}$ . Check yourself: $\angle BCD$ should measure $70^{\circ}$ , as opposite angles of a parallelogram are equal. Then measure the diagonals $AC$ and $BD$ .

ShowHide solution

Step 1. Draw $AB = 5$ cm. At $A$ measure $70^{\circ}$ , draw the line and mark $D$ on it, 5 cm from $A$ .
Step 2. All four sides are 5 cm. Set the compasses to 5 cm and draw arcs with centres $B$ and $D$ ; they cross at $A$ and at $C$ , the crossing on the other side of $BD$ from $A$ .
Step 3. Join $BC$ and $DC$ . Check $\angle BCD = 70^{\circ}$ , then join and measure $AC$ and $BD$ .

The construction

The diagonals, measured

$AC = 8.2$ cm (accurate if from 8.0 cm to 8.4 cm) and $BD = 5.7$ cm (accurate if from 5.5 cm to 5.9 cm).

Practice 4

Wei Ming wants to cut out a square card whose diagonals are 6 cm long. Construct the square $ABCD$ with the diagonal $AC = 6$ cm. Check yourself: $BD$ should measure 6 cm and $\angle ABC$ should be $90^{\circ}$ . Then measure the side $AB$ .

ShowHide solution

Step 1. The diagonals of a square are equal and bisect each other at right angles, so $BD = 6$ cm and the diagonals cross at the midpoint $O$ of each, with $OA = OB = OC = OD = 3$ cm.
Step 2. Draw $AC = 6$ cm and mark its midpoint $O$ , 3 cm from $A$ .
Step 3. With the set square and ruler, draw the line through $O$ perpendicular to $AC$ . Set the compasses to 3 cm and, with centre $O$ , draw arcs cutting it at $B$ and $D$ .
Step 4. Join $AB$ , $BC$ , $CD$ and $DA$ . Check $BD = 6$ cm and $\angle ABC = 90^{\circ}$ , then measure $AB$ .

$AB = 4.2$ cm (accurate if from 4.0 cm to 4.4 cm). Wei Ming's card has sides of about 4.2 cm.

Practice 5

Construct the rectangle $ABCD$ with $AB = 6$ cm and the diagonal $AC = 10$ cm. Check yourself: $AC$ should measure 10 cm and $\angle ABC$ should be $90^{\circ}$ . Then measure $BC$ .

ShowHide solution

Step 1. Draw $AB = 6$ cm. With the set square and ruler, draw the line perpendicular to $AB$ at $B$ .
Step 2. Set the compasses to 10 cm. With centre $A$ , draw an arc cutting the perpendicular at $C$ . (This is the right-angled triangle $ABC$ from its hypotenuse and one other side, as in Constructing Triangles.)
Step 3. Draw the line perpendicular to $AB$ at $A$ , and the line through $C$ parallel to $AB$ . They meet at $D$ .
Step 4. Check $AC = 10$ cm, then measure $BC$ .

$BC = 8.0$ cm (accurate if from 7.8 cm to 8.2 cm).

Practice 6

Priya is cutting out a card in the shape of a trapezium $PQRS$ with $PQ \parallel SR$ , $PQ = 9$ cm, $PS = 4$ cm, $\angle SPQ = 70^{\circ}$ and $\angle PQR = 55^{\circ}$ . Construct the trapezium. Check yourself: measure $\angle SPQ$ again. Then measure $SR$ and $QR$ .

ShowHide solution

Step 1. Draw $PQ = 9$ cm. At $P$ measure $70^{\circ}$ and mark $S$ , 4 cm from $P$ .
Step 2. At $Q$ measure $55^{\circ}$ and draw a long line on the same side of $PQ$ as $S$ .
Step 3. With the set square and ruler, draw the line through $S$ parallel to $PQ$ . It meets the line from $Q$ at $R$ .
Step 4. Check $\angle SPQ = 70^{\circ}$ , then measure $SR$ and $QR$ .

$SR = 5.0$ cm (accurate if from 4.8 cm to 5.2 cm) and $QR = 4.6$ cm (accurate if from 4.4 cm to 4.8 cm).

Practice 7

Mei Ling is making a paper kite $ABCD$ with $AB = AD$ and $CB = CD$ . Its diagonals are $AC = 8$ cm and $BD = 5$ cm, and they cross at $O$ with $AO = 2.5$ cm. Construct the kite. Check yourself: $BD$ should measure 5 cm. Then measure $AB$ , $CB$ and $\angle ABC$ .

ShowHide solution

Step 1. $AC$ joins the vertices where the equal sides meet, so it is the line of symmetry: it cuts $BD$ in half at right angles (prop. of kite). So $BO = OD = 2.5$ cm.
Step 2. Draw $AC = 8$ cm and mark $O$ on it, 2.5 cm from $A$ .
Step 3. With the set square and ruler, draw the line through $O$ perpendicular to $AC$ . Set the compasses to 2.5 cm and, with centre $O$ , draw arcs cutting it at $B$ and $D$ .
Step 4. Join $AB$ , $BC$ , $CD$ and $DA$ . Check $BD = 5$ cm, then measure $AB$ , $CB$ and $\angle ABC$ .

$AB = 3.5$ cm (accurate if from 3.3 cm to 3.7 cm), $CB = 6.0$ cm (accurate if from 5.8 cm to 6.2 cm) and $\angle ABC = 111^{\circ}$ (accurate if from $109^{\circ}$ to $113^{\circ}$ ).

Practice 8

Construct the quadrilateral $ABCD$ with $AB = 5$ cm, $BC = 6$ cm, $CD = 4$ cm, $DA = 5.5$ cm and the diagonal $BD = 7$ cm. Check yourself: $CD$ should measure 4 cm. Then measure the diagonal $AC$ .

ShowHide solution

Step 1. The diagonal $BD$ splits $ABCD$ into $\triangle ABD$ (sides 5 cm, 5.5 cm, 7 cm) and $\triangle BCD$ (sides 6 cm, 4 cm, 7 cm).
Step 2. Draw $AB = 5$ cm. Draw an arc of radius 5.5 cm with centre $A$ and an arc of radius 7 cm with centre $B$ ; they cross at $D$ . Join $AD$ and $BD$ .
Step 3. On the other side of $BD$ from $A$ , draw an arc of radius 6 cm with centre $B$ and an arc of radius 4 cm with centre $D$ ; they cross at $C$ . Join $BC$ and $DC$ .
Step 4. Check $CD = 4$ cm, then join $AC$ and measure it.

$AC = 7.5$ cm (accurate if from 7.3 cm to 7.7 cm).

Practice 9

Construct the quadrilateral $PQRS$ with $PQ = 7$ cm, $\angle SPQ = 80^{\circ}$ , $\angle PQR = 75^{\circ}$ , $PS = 4.5$ cm and $QR = 5$ cm. Check yourself: measure $PS$ again. Then measure $SR$ and $\angle PSR$ .

ShowHide solution

Step 1. Draw $PQ = 7$ cm.
Step 2. At $P$ measure $80^{\circ}$ and mark $S$ , 4.5 cm from $P$ . At $Q$ measure $75^{\circ}$ , on the same side of $PQ$ , and mark $R$ , 5 cm from $Q$ .
Step 3. Join $SR$ . Check $PS = 4.5$ cm, then measure $SR$ and $\angle PSR$ .

$SR = 4.9$ cm (accurate if from 4.7 cm to 5.1 cm) and $\angle PSR = 105^{\circ}$ (accurate if from $103^{\circ}$ to $107^{\circ}$ ).

Practice 10

Construct the quadrilateral $PQRS$ with $PQ = QR = RS = 5$ cm, $\angle PQR = 100^{\circ}$ and $\angle QRS = 80^{\circ}$ , with $P$ and $S$ on the same side of $QR$ . Check yourself: measure $\angle PQR$ again. Then measure $PS$ and $\angle QPS$ , and name the most specific special quadrilateral that $PQRS$ is.

ShowHide solution

Step 1. Draw $QP = 5$ cm. At $Q$ measure $\angle PQR = 100^{\circ}$ and mark $R$ , 5 cm from $Q$ .
Step 2. At $R$ measure $\angle QRS = 80^{\circ}$ , on the same side of $QR$ as $P$ , and mark $S$ , 5 cm from $R$ .
Step 3. Join $PS$ . Check $\angle PQR = 100^{\circ}$ , then measure $PS$ and $\angle QPS$ .

Measured: $PS = 5.0$ cm and $\angle QPS = 80^{\circ}$ .

$\angle PQR + \angle QRS = 100^{\circ} + 80^{\circ} = 180^{\circ}$ , so $QP \parallel RS$ (int. ∠s supp.). And $\angle QPS + \angle PQR = 80^{\circ} + 100^{\circ} = 180^{\circ}$ , so $PS \parallel QR$ (int. ∠s supp.). Both pairs of opposite sides are parallel, so $PQRS$ is a parallelogram.

All four sides are 5 cm, so it is a rhombus. Its angles are not right angles, so it is not a square.

$PS = 5.0$ cm (accurate if from 4.8 cm to 5.2 cm), $\angle QPS = 80^{\circ}$ (accurate if from $78^{\circ}$ to $82^{\circ}$ ), and $PQRS$ is a rhombus.

Key points

Sketch first, construct with instruments, leave the arcs showing, check yourself on a given length or angle, then measure (to 0.1 cm and to the nearest degree).
The perpendicular from $P$ to $AB$ : one edge of the set square's right angle along a ruler laid on $AB$ ; slide until the other edge passes through $P$ . $PN$ is the distance from $P$ to $AB$ .
The line through $P$ parallel to $AB$ : one edge of the set square along $AB$ , a ruler against another edge; slide along the ruler until the first edge passes through $P$ .
A parallelogram from two sides and the angle between them: the other two sides by parallel lines, or by arcs, since opposite sides are equal.
A rhombus, a square or a kite from its diagonals: draw one diagonal, the perpendicular through the crossing point, and mark the other diagonal with compasses.
A rectangle: perpendiculars at both ends of one side. A trapezium: the parallel side through the end of a slanting side.
Any quadrilateral: a diagonal makes two triangles; construct one, then the other on the other side of the diagonal. Four sides alone do not fix a quadrilateral.
Construct, then name: give the reason for each parallel pair, such as (int. ∠s supp.), then match the definitions for the most specific name.

Before you start

Perpendicular and Parallel Lines with a Set Square

Example 1: The perpendicular from a point to a line

The perpendicular from a point to a line

Drawing the line through PPP parallel to ABABAB

Why the line is parallel

Parallelograms, Rhombuses, Rectangles and Squares

Example 2: A parallelogram from two sides and the angle between them

Another way to find C: with compasses

Example 3: A rhombus from its diagonals

Example 4: A rectangle

Trapeziums and Kites

Example 5: A trapezium

Example 6: A kite from its diagonals

Any Quadrilateral: Two Triangles

Example 7: Four sides and a diagonal

Why four sides are not enough

Example 8: Construct, then name

Common mistakes

Practice

Practice 1

Practice 2

Practice 3

Practice 4

Practice 5

Practice 6

Practice 7

Practice 8

Practice 9

Practice 10

Key points

Drawing the line through $P$ parallel to $AB$